28th October 2013
Proof that India was the first user of zero as a MATHEMATICAL CONCEPT. A guest blog post by Dr Navin Singh
I've got a Facebook friend named Navin Singh. He often makes extremely well informed comments, although we may have occasional differences.
He provided some extremely competent comments during some discussion/ debate that had started in FB regarding the origin of the mathematical zero, which some suggested not first discovered in India. I requested him to compile his views so I could publish. Here is his write up (I've fixed a few spelling errors – but not touched the text), with my comments after the article.
Navin is an Orhopaedic and Spine surgeon, with a fellowship in arthroplasty, practising in Raipur, Chhattisgarh. Interested in science and music. he believe Indians should take more active interest in politics for our democracy to work, but say that "I am an armchair activist myself".
Zero: the nothing that is
Indians have always taken great pride in being the culture that invented zero. Unaware of our rich mathematical tradition, which inspired the Chinese, the Islamic World and the Europeans for centuries, our pysche has been fixated on the zero. For a nation with a dearth of heroes, grappling with a history riddled with foreign invasions and subjugation, this has been the proverbial last straw to hold on to, for us. But few recent studies, most notably a book by Kaplan(1999) which credits the Babylonians with inventing the zero by 3rd century BCE, and some others acknowledging the knowledge of zero by the Mayans circa 350 CE, conveniently predate the conservative estimate of the first textual evidence of zero in Jain texts by 5th century, and the stone inscription of the symbol, as a small circle, 'o' , as we use it even today, from a temple close to Gwalior dated to 9th century. The question thus arises is, whether zero was indeed India's contribution to the World, or India's contribution was…. just a, naught.
To look into this question, we have to make a quick recollection of the evolution of numeration in various cultures, and how it led to zero being invented and then…. 'discovered'. There is a generally accepted 'cardinal number' viewpoint which states that that counting started with practical needs like counting the cattle etc., i.e. to answer two basic questions- how many? and how much? There is an alternative 'ordinal' theory which says that humans learnt enumeration even earlier, for ritual needs, where participants of a rite were numbered, or for simply establishing a pecking order in a dominance hierarchy, i.e. to answer questions like – who gets to eat first, second etc.? or who comes first?
Earliest known record of counting, a baboon fibula marked with 29 notches, discovered in Africa, is from 35, 000 BCE. Very soon humans would have learnt to make groups or sets of numbers, to make the counting of larger numbers easy, as evidenced by discovery of a wolf tibia marked with 57 notches in groups of five (as we do even today) dated to 30, 000 BCE. Different cultures would have developed counting independently, but Seidenberg (1962) believes that it started at one place and then spread to the rest of the World. He points to the remarkable similarities in numbers of different cultures, like even numbers always being feminine and the odds, masculine (this distinction has been blurred in modern times, however).
Early humans used counters as pebbles or notches on stick or bones and they would have soon noted the convenience of using body parts for counting, and even today many cultures have names of body parts as numbers (like the word digits, which means fingers as well a numbers) . Fingers came in handy and the ten fingers gave most cultures a decimal system of enumeration, where numbers are grouped in sets of ten, or it has a base 10, and numbers are grouped as 10, 10 square, 10 cube and so on. But several cultures, like certain Mexican tribes had an octal number system with base 8, as they used the web spaces of fingers for counting, instead of fingers. Using thumbs for counting the segments or phalanges of other four digits gives one the count of 12 as each digit except thumb has three phalanges each. So some culture evolved a duo-decimal or dozenal system, where 12 make a dozen, 12 dozens – a gross, 12 grosses – a great gross. Using only one hand gave a base 5 number system, for some African tribes. In addition, there were cultures with hexagesimal (base 16), vigesimal (base 20, both hands and toes? :)) and sexagesimal (base 60) systems.
It is easy to see the superiority of the decimal system, its easier to find six time ten than it is to calculate six times a dozen. And its easier to count in fives than it is to count in sixes. Right? Far from it. Mathematically, there is nothing special about ten. The ease of decimal system is just an illusion. When we are mentally calculating six times ten as sixty, we are merely stating that six times ten are six sets of ten. Six times a dozen similarly means six sets of a dozen or 72 (which would look as 60 in modern notation, if we have a dozenal system). Counting by five is easy as it is half of ten, so four times five is four 'half tens' or two 'tens'. If we had adopted a dozenal system, we would be counting with sixes, with equal ease. But it is true that numbers systems with larger bases, like 20 or 60, necessitate greater number of symbols, multiplication tables become unwieldy, making mental arithmetic tedious. So number systems with smaller bases like 8, 10, 12 or 16 are more convenient.
There is very little to choose between these smaller bases as far as addition, subtraction and multiplication is considered, but division is easier, at least for the lay person, in bases 8, 12 and 16, compared to decimal system. This is because, the latter two have greater number of factors than ten. In addition, 16 and 8 are binary i.e. we can keep halving them till we get 1, and we won't encounter a fraction, whereas 10 can be halved but its quarter yields a fraction, 2.5. This convenience of base 16 counting is the reason why old Indian Rupee was made up of 16 annas. A dozenal system has the advantage that 12 can be divided into thirds while a third of 10 yields a recurring decimal fraction, 3.3333. Apart from this, all these systems are equally good for arithmetic, so there is no point giving up the decimal system for a dozenal system like the various dozenal societies of the World insist.
Continuing our discussion further, once humans learnt to divide numbers in small sets (of 10 from now on, for ease of comprehension), they developed an additive system. The sets of ten added up to make larger numbers as needed. They needed names (and later symbols, when script developed) for numbers from 1-9, 10, two to ten 10s , ten 'ten 10s' or 100 tens and so on (Ancient Indians outpaced everyone by light-years, they had name for all the additions of ten up to '140 tens' or 140 zeroes after 1 in modern notation) . Egyptians made hieroglyphics for every digit and then for every multiple of ten, Romans had symbols for one, five, ten, fifty, hundred, five hundred and thousand. So 1234, was written as MCCXXXIIII (or MCCXXXIV as we write today), in Roman script; one symbol of 1000, two of 100, three of 10 and four of 1 in Egyptian system – ten symbols each. It can be represented in mere four symbols in modern notation. Additive systems needed abacus or counters for addition and subtraction (though it can be done using pen and paper, adding or subtracting similar symbols), multiplication and division was very difficult, it entailed repeated addition or subtraction, respectively. Mental arithmetic was virtually impossible. No wonder Europeans, using the Roman numerals, made no headway in Maths (or Science) till they adopted Hindu-Arab numerals, albeit with great difficulty, by sixteenth century. Additive systems needed no zero, as the symbols could be placed anywhere, left or right, without altering the value of quantity they represent (though Roman numerals are written in an order of higher to lower values, by convention and so are some other numerals).
Additive systems gave way to hybrid systems (and perhaps many would have been hybrid systems from the outset), those that use both addition and multiplication of the base, to add multiples of the base number, to create larger numbers. When we speak or write numbers in English, we use a hybrid system. So 1234 is written as one thousand, two hundred and thirty four, each number is followed by its place value, so zero is still not needed. Many cultures, like Chinese and Indians, then developed a pure multiplicative positional system, a system where symbols for nine digits, 1-9 were arranged in an order of place values to denote all numbers, however big or small. The null quantity in any place value was denoted by empty space in Chinese counting rods or as an empty column in an Indian sand abacus, by first millenium CE. These cultures now needed a place-holder for that vacant column.
Babylonians had a multiplicative, sexagesimal place value system for a thousand years, without a place holder. They had only two symbols for numbers, wedges and crescents, where right column had units represented from 1-59, next place had 60s, and left column had 60 squares or 3600s. So, if we write their numbers in modern notation, 61 (60+1) would look like 11, but so would 3601 (60 square+ 0+1), in the absence of a zero. By third century, BCE they started placing two slanted wedges in empty columns or positions, so, 61 would look like 11 (if written in our notation) and 3601 would look like 101. But invention of zero as a placeholder was still not complete, as this placeholder was never used at the end. So while in modern numerals, 10, 10 square and 10 cube can be differentiated by the number of zeroes in the end as -10, 100, 1000; in Babylonian system 60 and 60 square (3600) look alike- as a single wedge mark, their value being determined as per the context. Secondly, Babylonians failed to grasp the concept of zero as a number and so did the Mayans, who did invent a placeholder for their complicated numerals by 350 CE, which had a base 20 but wasn't exactly vigesimal. Their numerals became inconsistent with the base value beyond third place, so their zero didn't achieve the operational usability for arithmetic. Their numerals found their application in their intricate calendars, which had 18 months of 20 days in an year. So their units place had 1-19 digits, next place value had 20s, and the next place, instead of 20 square or 400 had just 360 (18×20), in accordance with their calendars, so it could have had little application in exact sciences.
It was then, the genius of Indian mathematicians which invented a zero, as a placeholder, complete with a positional base notation of a convenient and popular base of 10, which put mathematics on its present trajectory. An example of Indian texts mentioning zero is as follows- Viya dambar akasasa yama rama veda which means- sky(0), atmosphere (0), space (0), void (0), primordial couple (2), rama (3), veda(4). Indian place value notation in verse form moves from units to higher place values, from left to right, in modern notation it means 4, 320, 000, a kalpa. The total number of syllables in Rig Veda, the earliest of the Vedas, conservatively dated to 11th century BCE, is believed to be 432, 000 which according to de Santillana and von Dechend (1969) shows that the concept of kalpa has existed since early vedic periods. In addition, in Rig Veda 4.58, the verses 2 and 3, speak of a revelation of a buffalo with four horns, three feet, two heads and seven hands, and McClain (1978) believes that this is an allusion to kalpa, i.e. 4, 3, 2 and seven 0s – 4, 320, 000, 000, which, he argues, shows the knowledge of a positional number system at the time of composition of this verse.
The credit of invention of zero can be debated but when it comes to the discovery of zero as a number, the credit, without a doubt, goes to Brahmagupta in seventh century (the person who uses zero for the first time in an arithmetic operation like 10+1=11, i.e. as a placeholder, can be termed an inventor of the symbol zero, while one who uses it for an operation like 0+1=1, has discovered the concept of zero as a number). So while ancient Greeks wondered, how can nothing be something, a few centuries later, Brahmagupta set out to formalize arithmetic operations using zero. His rules for zero can be summed up as, n+0=n, 0+0=0, 0+(-n)= -n, 0+(+n)=n, 0-(-n)= n, 0-(+n)= -n, n×0=0, 0×0=0, 0/0=0 and n/0=0 (Colebrook trans. 1817). The last two, concerned with divisions involving zero are incorrect, five centuries later Bhaskara would lament that despite the passage of time he is still struggling to explain division by zero. He tried to get around this problem by stating n/0= infinity (a lot of maths teachers in India would give the same answer, perhaps a testimony to the enduring popularity of the teachings of Bhaskaracharya). According to present understanding, division by zero is left undefined. Bhaskara, however stated other operations like 0 square= 0 and square root of 0=0, correctly.
The establishment of zero as a number made mathematization of natural sciences possible, and the knowledge of zero was crucial in several fields in mathematics (e.g. the infinitesimal method, Dirac delta function, binary number system, binary metrics), physics (e.g. origin of the universe, laws of thermodynamics, zero temperature), chemistry (e.g. quantum numbers, chemical equilibrium in reversible thermodynamics) and biology (e.g. biological clock, origin of life) (Pogliani et al 1998).
Such is the enduring legacy of zero and Indian mathematics.
I agree that the use of zero as a mathematical entity seems to have been CLEARLY Indian. The decimal system, of course, was Indian. Most KEY mathematical advances – critical to the growth of science and measurement – came from India.
Whether India is contributing equally well today is highly questionable, though. The nature of our governance has seen the exodus of genius from India to other nations. And what remains behind in India is left to struggle with the basics of life, making India a laggard in mathematics and other disciplines today (at least as judged by PISA and other tests).